Find grammars for $=\{a,b\}$ that generate the sets of

$($a$)$ all strings with exactly two $a\text{'}$s$.$

$($b$)$ all strings with at least two $a\text{'}$s$.$

$($c$)$ all strings with no more than three $a\text{'}$s$.$

$($d$)$ all strings with at least three $a\text{'}$s$.$

$($e$)$ all strings that start with a and end with $b.$

$($f$)$ all strings with an even number of $b\text{'}$s$.$

In each case, give convincing arguments that the grammar you indeed generate the indicated language.